A115217 Diagonal sums of "correlation triangle" for 2^n.

1, 2, 6, 13, 30, 62, 133, 270, 558, 1125, 2286, 4590, 9253, 18542, 37230, 74533, 149358, 298862, 598309, 1196910, 2394990, 4790565, 9583470, 19168110, 38340901, 76684142, 153377646, 306759973, 613538670, 1227086702, 2454210853

Offset: 0

Author

Paul Barry, Jan 16 2006

Keywords

• easy
• nonn
• changed

Comments

Diagonal sums of number triangle A003983.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-2,-2,4).

Crossrefs

Cf. A003983.

Programs

Mathematica

LinearRecurrence[{2,2,-3,-2,-2,4},{1,2,6,13,30,62},40] (* Harvey P. Dale, Oct 18 2021 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} [j<=k]*2^(k-j)*[j<=n-2k]*2^(n-2k-j).

From Paul Barry, Jan 18 2006: (Start)

G.f.: 1/((1-2*x)*(1-2*x^2)*(1-x^3)).

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4) - 2*a(n-5) + 4*a(n-6). (End)

E.g.f.: (exp(x)*(7 + 48*exp(x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2) - 36*cosh(sqrt(2)*x) - 30*sqrt(2)*sinh(sqrt(2)*x))/21. - Stefano Spezia, Aug 28 2025